Abstract
A discrimination problem consists of N linearly independent pure quantum states Φ={φ_i} and the corresponding occurrence probabilities η={η_i}. To any such problem we associate, up to a permutation over the probabilities {η_i}, a unique pair of density matrices ρ__T and η_p defined on the N-dimensional Hilbert space H_N. The first one, ρ__T, provides a new parametrization of a generic full-rank density matrix in terms of the parameters of the discrimination problem, i.e. the mutual overlaps γ_ij=φ_iφ_j and the occurrence probabilities {η_i}. The second one is defined as a diagonal density matrix η_p with the diagonal entries given by the probabilities {η_i} with the ordering induced by the permutation p of the probabilities. ρ__T and η_p capture information about the quantum and classical versions of the discrimination problem, respectively. In this sense, when the set Φ can be discriminated unambiguously with probability one, i.e. when the states to be discriminated are mutually orthogonal and can be distinguished by a classical observer, then ρ__T→ η_p. Moreover, if the set lacks its independency and cannot be discriminated anymore the distinguishability of the pair, measured by the fidelity F(ρ__T, η_p), becomes minimum. This enables one to associate to each discrimination problem a measure of discriminability defined by the fidelity F(ρ__T, η_p). This quantity, has the advantage of being easy to calculate and in this respect it can find useful applications in estimating the extent to which the set is discriminable.